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The '''vulture family''' of [[temperament]]s [[tempering out|tempers out]] the [[vulture comma]] ({{monzo|legend=1| 24 -21 4 }}, [[ratio]]: 10485760000/10460353203), a small [[5-limit]] comma of 4.2 [[cent]]s.
{{Technical data page}}
 
The '''vulture family''' of [[temperament]]s [[tempering out|tempers out]] the [[vulture comma]] ({{monzo|legend=1| 24 -21 4 }}, [[ratio]]: 10 485 760 000 / 10 460 353 203), a small [[5-limit]] comma of 4.2 [[cent]]s that is the amount by which a stack of four [[syntonic comma]]s falls short of the [[256/243]] Pythagorean limma. As their defining feature, vulture temperaments split the interval [[3/1]] into four segments (identified in the 5-limit as [[320/243]]).
Temperaments discussed elsewhere include [[Landscape microtemperaments #Terture|terture]]. Considered below are septimal vulture, buzzard, condor, eagle, and turkey.  


== Vulture ==
== Vulture ==
The generator of the vulture temperament is a grave fourth of [[320/243]], that is, a [[4/3|perfect fourth]] minus a [[81/80|syntonic comma]]. Four of these make a [[3/1|perfect twelfth]]. Its [[ploidacot]] is alpha-tetracot.  
The generator of the vulture temperament is a grave fourth of [[320/243]], that is, a [[4/3|perfect fourth]] minus a [[81/80|syntonic comma]]. Four of these make a [[3/1|perfect twelfth]]. Its [[ploidacot]] is alpha-tetracot. It is a member of the [[syntonic–diatonic equivalence continuum]] with {{nowrap| ''n'' {{=}} 4 }}, so it equates a [[256/243|Pythagorean limma]] with a stack of four syntonic commas. It is also in the [[schismic–Mercator equivalence continuum]] with {{nowrap|''n'' {{=}} 4}}, so unless [[53edo]] is used as a tuning, the [[schisma]] is always observed.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 17: Line 16:


{{Mapping|legend=1| 1 0 -6 | 0 4 21 }}
{{Mapping|legend=1| 1 0 -6 | 0 4 21 }}
: mapping generators: ~2, ~320/243
: mapping generators: ~2, ~320/243


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~320/243 = 475.5351
* [[WE]]: ~2 = 1199.9430{{c}}, ~320/243 = 475.5200{{c}}
: [[error map]]: {{val| 0.0000 +0.1855 -0.0758 }}
: [[error map]]: {{val| -0.057 +0.125 -0.051 }}
* [[POTE]]: ~2 = 1200.000, ~320/243 = 475.5426
* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5396{{c}}
: error map: {{val| 0.0000 +0.2154 +0.0811 }}
: error map: {{val| 0.000 +0.203 +0.018 }}


{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b }}
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 323, 2531, 2854b, 3177b, …, 4469b }}


[[Badness]] (Smith): 0.041431
[[Badness]] (Sintel): 0.972


Badness (Dirichlet): 0.972
=== Overview to extensions ===
Temperaments discussed elsewhere include [[Buzzardsmic clan #Buzzard|buzzard]]. Considered below are septimal vulture, terture, condor, eagle, and turkey.


=== 2.3.5.19 ===
== Septimal vulture ==
It can be observed that the generator of vulture is very close to [[25/19]]; this corresponds to tempering out [[1216/1215]] = ([[19/15]])/([[9/8|18/16]])<sup>2</sup> = S16/S18. It results in a surprising decrease in Dirichlet badness, and (up to [[octave equivalence]]) finds [[19/16]] at 41 generators so that [[19/10]] is found at 20 generators, [[38/27]] is found at 18, [[19/15]] is found at 16 (as 3 is found at 4) and 76/45 is found at 12 so that it's equated with [[27/16]], which is tuned slightly sharp, as 76/45 is 1216/1215 above it. As a result of the interpretation of 1 gen as ~25/19, the 3 gen interval of ~226.6{{cent}} is interpreted as ([[3/2]])/([[25/19]]) = [[~]][[57/50]] which is tuned ~0.2{{cent}} flat. (Interpreting this interval as a damaged [[~]][[8/7]] leads to [[#Buzzard]].) Note that unless you are fine with the low accuracy* tuning offered by [[53edo]], you cannot temper out the [[schisma]], nor can you equate 32/27 with 19/16 or 24/19 with 19/15, meaning both the schisma and [[513/512]][[~]][[361/360]] (resp.) are observed. {{nowrap|* Compared}} to what this microtemperament is capable of. This means that the step size of [[270edo]] is especially ideal, being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Also note that 164 - 53 = 53 + 58 = [[111edo]] is a possible tuning which doesn't appear in the optimal ET sequence because it's less accurate than 53edo on the 2.3.5.19 subgroup.
Septimal vulture can be described as the {{nowrap| 53 & 270 }} microtemperament, tempering out the [[ragisma]], 4375/4374 and the [[garischisma]], 33554432/33480783 ({{monzo| 25 -14 0 -1 }}) aside from the vulture comma. [[270edo]] is an excellent tuning for this temperament, with generator 107\270. Other compatible tunings include [[217edo]] and [[323edo]]. The harmonic 7 is found at -14 fifths or {{nowrap| (-14) × 4 {{=}} -56 }} generator steps, so the smallest [[mos scale]] that includes it is the 58-note one, though for larger scope of harmony, you could try the 111- or 164-note one. For a much simpler mapping of 7 at the cost of higher error, you could try [[Buzzardsmic clan #Septimal buzzard|buzzard]].  
 
Subgroup: 2.3.5.19
 
Commas: 1216/1215, 64000000/63950067
 
{{Mapping|legend=1| 1 0 -6 -12 | 0 4 21 41 }}
 
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~25/19 = 475.542
: [[error map]]: {{val| 0.000 +0.214 +0.075 -0.278  }}


{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 593, 863 }}
It can be extended to the 11-limit by identifying a stack of four [[5/4]]'s as [[11/9]], tempering out [[5632/5625]], and to the 13-limit by identifying the hemitwelfth as [[26/15]], tempering out [[676/675]]. Furthermore, the generator of vulture is very close to [[25/19]]; a stack of three generator steps octave-reduced thus represents its fifth complement, [[57/50]]. This corresponds to tempering out [[1216/1215]] with the effect of equating the schisma with [[513/512]] and [[361/360]] in addition to many 11- and 13-limit commas. 270edo remains an excellent tuning in all cases.  
 
Badness (Dirichlet): 0.232
 
== Septimal vulture ==
Septimal vulture can be described as the {{nowrap| 53 & 270 }} microtemperament, tempering out the [[ragisma]], 4375/4374 and the [[garischisma]], 33554432/33480783 ({{monzo| 25 -14 0 -1 }}) aside from the vulture comma. [[270edo]] is a good tuning for this temperament, with generator 107\270. The harmonic 7 is found at -14 fifths or {{nowrap| (-14) × 4 {{=}} -56 }} generator steps, so that the smallest mos scale that includes it is the 58-note one, though for larger scope for harmony, you could try the 111- or 164-note one. For a much simpler mapping of 7 at the cost of higher error, you could try [[#Buzzard|buzzard]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 56: Line 41:


{{Mapping|legend=1| 1 0 -6 25 | 0 4 21 -56 }}
{{Mapping|legend=1| 1 0 -6 25 | 0 4 21 -56 }}
{{Multival|legend=1| 4 21 -56 24 -100 -189 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~320/243 = 475.5528
* [[WE]]: ~2 = 1199.9050{{c}}, ~320/243 = 475.5135{{c}}
: [[error map]]: {{val| 0.0000 +0.2561 +0.2945 +0.2188 }}
: [[error map]]: {{val| -0.095 +0.099 +0.039 +0.044 }}
* [[POTE]]: ~2 = 1200.0000, ~320/243 = 475.5511
* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/243 = 475.5515{{c}}
: error map: {{val| 0.0000 +0.2495 +0.2601 +0.3106 }}
: error map: {{val| 0.000 +0.251 +0.267 +0.292 }}


{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 593, 863, 1133 }}
{{Optimal ET sequence|legend=1| 53, 164, 217, 270, 593, 863, 1133, 1996d }}


[[Badness]] (Smith): 0.036985
[[Badness]] (Sintel): 0.936


=== 11-limit ===
=== 11-limit ===
Line 77: Line 60:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~320/243 = 475.5558
* WE: ~2 = 1199.9392{{c}}, ~320/243 = 475.5326{{c}}
* POTE: ~2 = 1200.0000, ~320/243 = 475.5567
* CWE: ~2 = 1200.0000{{c}}, ~320/243 = 475.5655{{c}}


{{Optimal ET sequence|legend=0| 53, 217, 270, 2107c, 2377bc }}
{{Optimal ET sequence|legend=0| 53, 217, 270, 2107c, 2377bc }}


Badness (Smith): 0.031907
Badness (Sintel): 1.05


==== 13-limit ====
==== 13-limit ====
Line 92: Line 75:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~320/243 = 475.5566
* WE: ~2 = 1199.9695{{c}}, ~154/117 = 475.5451{{c}}
* POTE: ~2 = 1200.0000, ~320/243 = 475.5572
* CWE: ~2 = 1200.0000{{c}}, ~154/117 = 475.5571{{c}}


{{Optimal ET sequence|legend=0| 53, 217, 270 }}
{{Optimal ET sequence|legend=0| 53, 217, 270 }}


Badness (Smith): 0.018758
Badness (Sintel): 0.775


==== 17-limit ====
==== 2.3.5.7.11.13.19 subgroup ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.19


Comma list: 676/675, 936/935, 1001/1000, 1225/1224, 4096/4095
Comma list: 676/675, 1001/1000, 1216/1215, 1540/1539, 1729/1728


Mapping: {{mapping| 1 0 -6 25 -33 -7 35 | 0 4 21 -56 92 27 -78 }}
Mapping: {{mapping| 1 0 -6 25 -33 -7 -12 | 0 4 21 -56 92 27 41 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~112/85 = 475.5613
* WE: ~2 = 1199.9636{{c}}, ~25/19 = 475.5426{{c}}
* POTE: ~2 = 1200.0000, ~112/85 = 475.5617
* CWE: ~2 = 1200.0000{{c}}, ~25/19 = 475.5569{{c}}


{{Optimal ET sequence|legend=0| 53, 217, 270, 487, 757g }}
{{Optimal ET sequence|legend=0| 53, 217, 270 }}
 
Badness (Smith): 0.020103
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 676/675, 936/935, 1001/1000, 1216/1215, 1225/1224, 1540/1539
 
Mapping: {{mapping| 1 0 -6 25 -33 -7 35 -12 | 0 4 21 -56 92 27 -78 41 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~25/19 = 475.5606
* POTE: ~2 = 1200.0000, , ~25/19 = 475.5615
 
{{Optimal ET sequence|legend=0| 53, 217, 270, 487, 757g }}


Badness (Smith): 0.013850
Badness (Sintel): 0.579


=== Semivulture ===
=== Semivulture ===
Line 135: Line 103:


Mapping: {{mapping| 2 0 -12 50 41 | 0 4 21 -56 -43 }}
Mapping: {{mapping| 2 0 -12 50 41 | 0 4 21 -56 -43 }}
: mapping generators: ~99/70, ~320/243
: mapping generators: ~99/70, ~320/243


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.0000, ~320/243 = 475.5523
* WE: ~99/70 = 599.9594{{c}}, ~320/243 = 475.5174{{c}}
* POTE: ~99/70 = 600.0000, ~320/243 = 475.5496
* CWE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5501{{c}}


{{Optimal ET sequence|legend=0| 106, 164, 270, 916, 1186, 1456 }}
{{Optimal ET sequence|legend=0| 106, 164, 270, 916, 1186, 1456 }}


Badness (Smith): 0.040799
Badness (Sintel): 1.35


==== 13-limit ====
==== 13-limit ====
Line 154: Line 121:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.0000, ~320/243 = 475.5540
* WE: ~99/70 = 599.9859{{c}}, ~320/243 = 475.5423{{c}}
* POTE: ~99/70 = 600.0000, ~320/243 = 475.553
* CWE: ~99/70 = 600.0000{{c}}, ~320/243 = 475.5536{{c}}


{{Optimal ET sequence|legend=0| 106, 164, 270 }}
{{Optimal ET sequence|legend=0| 106, 164, 270 }}


Badness (Smith): 0.035458
Badness (Sintel): 1.47


== Buzzard ==
== Terture ==
{{Main| Buzzard }}
Named by [[Xenllium]] in 2021, terture tempers out 250047/250000, the [[landscape comma]], and may be described as the {{nowrap| 111 & 159 }} temperament, with a [[ploidacot]] signature of triploid gamma-tetracot.  
{{See also| No-fives subgroup temperaments #Buzzard }}
 
Buzzard is the main extension to vulture of practical interest, finding prime 7 at only 3 generators down so that the generator is interpreted as a sharp ~[[21/16]], but is more of a full 13-limit system in its own right. It is most naturally described as 53 & 58 (though [[48edo]] is an interesting higher-damage tuning of it for some purposes). As one might expect, 111edo is a great tuning for it. [[mos scale]]s of 3, 5, 8, 13, 18, 23, 28, 33, 38, 43, 48 or 53 notes are available.
 
Its 13-limit [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]], [[847/845|S11/S13]], [[676/675|S13/S15]]}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] × [[81/80|S9]]. [[Hemifamity]] leverages it by splitting [[36/35]] into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]]. Then the vanish of the orwellisma implies [[49/48]], the large septimal diesis, is equated with 36/35, so 49/48 is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is [[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in the full 13-limit.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 5120/5103
[[Comma list]]: 250047/250000, 359661568/358722675
 
{{Mapping|legend=1| 1 0 -6 4 | 0 4 21 -3 }}


{{Multival|legend=1| 4 21 -3 24 -16 -66 }}
{{Mapping|legend=1| 3 0 -18 -32 | 0 4 21 34 }}
: mapping generators: ~63/50, ~320/243


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~21/16 = 475.555
* [[WE]]: ~63/50 = 399.9723{{c}}, ~320/243 = 475.5221{{c}} (~392/375 = 75.5499{{c}})
: [[error map]]: {{val| 0.000 +0.263 +0.333 +4.510 }}
: [[error map]]: {{val| -0.083 +0.134 +0.151 -0.185 }}
* [[POTE]]: ~2 = 1200.000, ~21/16 = 475.636
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5519{{c}} (~392/375 = 75.5519{{c}})
: error map: {{val| 0.000 +0.589 +2.045 +4.266 }}
: error map: {{val| 0.000 +0.253 +0.276 -0.061 }}


{{Optimal ET sequence|legend=1| 5, 48, 53, 111, 164d, 275d }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}


[[Badness]] (Smith): 0.047963
[[Badness]] (Sintel): 2.21


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 540/539, 5120/5103
Comma list: 3025/3024, 19712/19683, 102487/102400
 
Mapping: {{mapping| 1 0 -6 4 -12 | 0 4 21 -3 39 }}


Wedgie: {{multival| 4 21 -3 39 24 -16 48 -66 18 120 }}
Mapping: {{mapping| 3 0 -18 -32 8 | 0 4 21 34 2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/16 = 475.625
* WE: ~63/50 = 399.9902{{c}}, ~320/243 = 475.5383{{c}} (~392/375 = 75.5481{{c}})
* POTE: ~2 = 1200.000, ~21/16 = 475.700
* CWE: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5490{{c}} (~392/375 = 75.5490{{c}})


{{Optimal ET sequence|legend=0| 53, 58, 111, 280cd }}
{{Optimal ET sequence|legend=0| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}


Badness (Smith): 0.034484
Badness (Sintel): 0.969


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 351/350, 540/539, 676/675
Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976


Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }}
Mapping: {{mapping| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}
 
Wedgie: {{multival| 4 21 -3 39 27 24 -16 48 28 -66 18 -15 120 87 -51 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/16 = 475.615
* WE: ~63/50 = 399.9958{{c}}, ~154/117 = 475.5485{{c}} (~117/112 = 75.5527{{c}})
* POTE: ~2 = 1200.000, ~21/16 = 475.697
* CWE: ~63/50 = 400.0000{{c}}, ~154/117 = 475.5531{{c}} (~117/112 = 75.5531{{c}})


{{Optimal ET sequence|legend=0| 53, 58, 111, 280cdf }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness (Smith): 0.018842
Badness (Sintel): 0.771


==== 17-limit ====
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913


Mapping: {{mapping| 1 0 -6 4 -12 -7 14 | 0 4 21 -3 39 27 -25 }}
Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 | 0 4 21 34 2 27 12 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/16 = 475.638
* WE: ~34/27 = 399.9664{{c}}, ~112/85 = 475.5198{{c}} (~117/112 = 75.5534{{c}})
* POTE: ~2 = 1200.000, ~21/16 = 475.692
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~117/112 = 75.5568{{c}})


{{Optimal ET sequence|legend=0| 53, 58, 111 }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness (Smith): 0.018403
Badness (Sintel): 0.953


==== 19-limit ====
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615


Mapping: {{mapping| 1 0 -6 4 -12 -7 14 -12 | 0 4 21 -3 39 27 -25 41 }}
Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 | 0 4 21 34 2 27 12 41 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/16 = 475.617
* WE: ~34/27 = 399.9665{{c}}, ~112/85 = 475.5198{{c}} (~95/91 = 75.5533{{c}})
* POTE: ~2 = 1200.000, ~21/16 = 475.679
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~95/91 = 75.5568{{c}})


{{Optimal ET sequence|legend=0| 53, 58h, 111 }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness (Smith): 0.015649
Badness (Sintel): 0.846


=== Buteo ===
=== 23-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 99/98, 385/384, 2200/2187
Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215


Mapping: {{mapping| 1 0 -6 4 9 | 0 4 21 -3 -14 }}
Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 10 | 0 4 21 34 2 27 12 41 3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/16 = 475.454
* WE: ~34/27 = 400.0026{{c}}, ~112/85 = 475.5510{{c}} (~24/23 = 75.5485{{c}})
* POTE: ~2 = 1200.000, ~21/16 = 475.436
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5482{{c}} (~24/23 = 75.5482{{c}})


{{Optimal ET sequence|legend=0| 5, 48, 53 }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness (Smith): 0.060238
Badness (Sintel): 1.07


==== 13-limit ====
== Condor ==
Subgroup: 2.3.5.7.11.13
Condor tempers out [[10976/10935]] and may be described as the {{nowrap| 58 & 159 }} temperament. The generator represents the [[112/81|septimal diminished fifth (112/81)]], and three minus an octave make vulture's generator of ~320/243. The ploidacot for this temperament is epsilon-dodecacot. [[217edo]] is an excellent tuning for this temperament.  
 
Comma list: 99/98, 275/273, 385/384, 572/567
 
Mapping: {{mapping| 1 0 -6 4 9 -7 | 0 4 21 -3 -14 27 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~21/16 = 475.495
* POTE: ~2 = 1200.000, ~21/16 = 475.464


{{Optimal ET sequence|legend=0| 5, 48f, 53 }}
Badness (Smith): 0.039854
== Condor ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 40353607/40000000
[[Comma list]]: 10976/10935, 40353607/40000000


{{Mapping|legend=1| 1 8 36 29 | 0 -12 -63 -49 }}
{{Mapping|legend=1| 1 -4 -27 -20 | 0 12 63 49 }}
: mapping generators: ~2, ~112/81


{{Multival|legend=1| 12 63 49 72 44 -63 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.0142{{c}}, ~112/81 = 558.5276{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 641.4791
: [[error map]]: {{val| +0.014 +0.319 +0.539 -1.260 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~112/81 = 558.5212{{c}}
: error map: {{val| 0.000 +0.300 +0.523 -1.287 }}


{{Optimal ET sequence|legend=1| 58, 159, 217 }}
{{Optimal ET sequence|legend=1| 58, 159, 217 }}


[[Badness]]: 0.154715
[[Badness]] (Sintel): 3.92


=== 11-limit ===
=== 11-limit ===
Line 301: Line 248:
Comma list: 441/440, 4000/3993, 10976/10935
Comma list: 441/440, 4000/3993, 10976/10935


Mapping: {{mapping| 1 8 36 29 35 | 0 -12 -63 -49 -59 }}
Mapping: {{mapping| 1 -4 -27 -20 -24 | 0 12 63 49 59 }}


Optimal tuning (POTE): ~2 = 1\1, 81/56 = 641.4822
Optimal tunings:
* WE: ~2 = 1199.9730{{c}}, ~112/81 = 558.5052{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/81 = 558.5173{{c}}


{{Optimal ET sequence|legend=1| 58, 101cd, 159, 217 }}
{{Optimal ET sequence|legend=0| 58, 101cd, 159, 217, 376d }}


Badness: 0.048401
Badness (Sintel): 1.60


=== 13-limit ===
=== 13-limit ===
Line 314: Line 263:
Comma list: 364/363, 441/440, 676/675, 10976/10935
Comma list: 364/363, 441/440, 676/675, 10976/10935


Mapping: {{mapping| 1 8 36 29 35 47 | 0 -12 -63 -49 -59 -81 }}
Mapping: {{mapping| 1 -4 -27 -20 -24 -34 | 0 12 63 49 59 81 }}


Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4797
Optimal tunings:
* WE: ~2 = 1199.9649{{c}}, ~112/81 = 558.5040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/81 = 558.5197{{c}}


{{Optimal ET sequence|legend=1| 58, 159, 217 }}
{{Optimal ET sequence|legend=0| 58, 159, 217 }}


Badness: 0.025469
Badness (Sintel): 1.05


=== 17-limit ===
=== 17-limit ===
Line 327: Line 278:
Comma list: 364/363, 441/440, 595/594, 676/675, 8624/8619
Comma list: 364/363, 441/440, 595/594, 676/675, 8624/8619


Mapping: {{mapping| 1 8 36 29 35 47 -5 | 0 -12 -63 -49 -59 -81 17 }}
Mapping: {{mapping| 1 -4 -27 -20 -24 -34 12 | 0 12 63 49 59 81 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 641.4794
Optimal tunings:
* WE: ~2 = 1199.9594{{c}}, ~112/81 = 558.5017{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/81 = 558.5202{{c}}


{{Optimal ET sequence|legend=1| 58, 159, 217 }}
{{Optimal ET sequence|legend=0| 58, 159, 217 }}


Badness: 0.021984
Badness (Sintel): 1.12


== Eagle ==
== Eagle ==
Eagle tempers out [[2401/2400]] and may be described as the {{nowrap| 58 & 270 }} temperament. It has a semi-octave period and a generator of ~28/27, four of which make a hemifourth which may be identified with 15/13, and two of those make a perfect fourth; its ploidacot thus is diploid wau-octacot. Compatible tunings include [[212edo]], [[270edo]], and [[328edo]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 341: Line 296:


{{Mapping|legend=1| 2 4 9 8 | 0 -8 -42 -23 }}
{{Mapping|legend=1| 2 4 9 8 | 0 -8 -42 -23 }}
: mapping generators: ~177147/125440, ~28/27
: mapping generators: ~177147/125440, ~28/27


{{Multival|legend=1|16 84 46 96 28 -129}}
[[Optimal tuning]]s:
 
* [[WE]]: ~177147/125440 = 599.9818{{c}}, ~28/27 = 62.2266{{c}}
[[Optimal tuning]] ([[POTE]]): ~177147/125440 = 1\2, ~28/27 = 62.229
: [[error map]]: {{val| -0.036 +0.159 +0.004 -0.184 }}
* [[CWE]]: ~177147/125440 = 600.0000{{c}}, ~28/27 = 62.2295{{c}}
: error map: {{val| 0.000 +0.209 +0.046 -0.105 }}


{{Optimal ET sequence|legend=1| 58, 154c, 212, 270, 752, 1022, 1292, 2854b }}
{{Optimal ET sequence|legend=1| 58, 154c, 212, 270, 752, 1022, 1292, 2854b }}


[[Badness]]: 0.059498
[[Badness]] (Sintel): 1.51


=== 11-limit ===
=== 11-limit ===
Line 359: Line 315:
Mapping: {{mapping| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}
Mapping: {{mapping| 2 4 9 8 12 | 0 -8 -42 -23 -49 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/27 = 62.224
Optimal tunings:
* WE: ~99/70 = 599.9796{{c}} ~28/27 = 62.2218{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.2251{{c}}


{{Optimal ET sequence|legend=1| 58, 154ce, 212, 270 }}
{{Optimal ET sequence|legend=0| 58, 154ce, 212, 270 }}


Badness: 0.024885
Badness (Sintel): 0.823


=== 13-limit ===
=== 13-limit ===
Line 372: Line 330:
Mapping: {{mapping| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 }}
Mapping: {{mapping| 2 4 9 8 12 13 | 0 -8 -42 -23 -49 -54 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/27 = 62.220
Optimal tunings:
* WE: ~99/70 = 599.9763{{c}} ~28/27 = 62.2174{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.2211{{c}}


{{Optimal ET sequence|legend=1| 58, 154cef, 212, 270 }}
{{Optimal ET sequence|legend=0| 58, 154cef, 212, 270 }}


Badness: 0.016282
Badness (Sintel): 0.673


== Turkey ==
== Turkey ==
Named by [[Xenllium]] in 2021, turkey may be described as the {{nowrap| 212 & 217 }} temperament. It is generated by a fifth sharp of just, close to 3\5 but on the flat side thereof, which can be interpreted as [[50/33]] in the 11-limit. Sixteen generators minus nine octaves make a perfect fifth; its ploidacot is thus theta-16-cot. [[429edo]] may be recommended as a tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4802000/4782969, 5250987/5242880
[[Comma list]]: 4802000/4782969, 5250987/5242880


{{Mapping|legend=1| 1 8 36 0 | 0 -16 -84 7 }}
{{Mapping|legend=1| 1 -8 -48 7 | 0 16 84 -7 }}
: mapping generators: ~2, ~3592/1715


{{Multival|legend=1|16 84 -7 96 -56 -252}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1147{{c}}, ~3592/1715 = 718.9483{{c}}
: [[error map]]: {{val| +0.115 +0.300 -0.161 -0.661 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3592/1715 = 718.8806{{c}}
: error map: {{val| 0.000 +0.134 -0.345 -0.990 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1715/1296 = 481.120
{{Optimal ET sequence|legend=1| 212, 429, 1070d }}


{{Optimal ET sequence|legend=1| 5, 207c, 212, 429 }}
[[Badness]] (Sintel): 5.34
 
[[Badness]]: 0.210964


=== 11-limit ===
=== 11-limit ===
Line 398: Line 363:
Comma list: 19712/19683, 42875/42768, 160083/160000
Comma list: 19712/19683, 42875/42768, 160083/160000


Mapping: {{mapping| 1 8 36 0 64 | 0 -16 -84 7 -151 }}
Mapping: {{mapping| 1 -8 -48 7 -87 | 0 16 84 -7 151 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.120
Optimal tunings:
* WE: ~2 = 1200.1131{{c}} ~50/33 = 718.9478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/33 = 718.8808{{c}}


{{Optimal ET sequence|legend=1| 212, 429 }}
{{Optimal ET sequence|legend=0| 212, 429 }}


Badness: 0.079694
Badness (Sintel): 2.63


=== 13-limit ===
=== 13-limit ===
Line 411: Line 378:
Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104
Comma list: 676/675, 1001/1000, 19712/19683, 31213/31104


Mapping: {{mapping| 1 8 36 0 64 47 | 0 -16 -84 7 -151 -108 }}
Mapping: {{mapping| 1 -8 -48 7 -87 -61 | 0 16 84 -7 151 108 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/25 = 481.118
Optimal tunings:
* WE: ~2 = 1200.1324{{c}} ~50/33 = 718.9608{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/33 = 718.8825{{c}}


{{Optimal ET sequence|legend=1| 212, 217, 429 }}
{{Optimal ET sequence|legend=0| 212, 217, 429 }}


Badness: 0.043787
Badness (Sintel): 1.81


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Vulture family| ]] <!-- main article -->
[[Category:Vulture family| ]] <!-- main article  
[[Category:Vulture| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Vulture_family"